I would like to compute the Area Under the Curve defined by a set of experimental values. I created a function to calculate an aproximation of the AUC using the Simpson's rule as I saw in this post. However, the function only works when it receives a vector of odd length. How can I modify the code to add the area of the last trapezoid when the input vector has an even length.
AUC <- function(x, h=1){
# AUC function computes the Area Under the Curve of a time serie using
# the Simpson's Rule (numerical method).
# https://link.springer.com/chapter/10.1007/978-1-4612-4974-0_26
# Arguments
# x: (vector) time serie values
# h: (int) temporal resolution of the time serie. default h=1
n = length(x)-1
xValues = seq(from=1, to=n, by=2)
sum <- list()
for(i in 1:length(xValues)){
n_sub <- xValues[[i]]-1
n <- xValues[[i]]
n_add <- xValues[[i]]+1
v1 <- x[[n_sub+1]]
v2 <- x[[n+1]]
v3 <- x[[n_add+1]]
s <- (h/3)*(v1+4*v2+v3)
sum <- append(sum, s)
}
sum <- unlist(sum)
auc <- sum(sum)
return(auc)
}
Here a data example:
smoothed = c(0.3,0.317,0.379,0.452,0.519,0.573,0.61,0.629,0.628,0.613,0.587,0.556,0.521,
0.485,0.448,0.411,0.363,0.317,0.273,0.227,0.185,0.148,0.12,0.103,0.093,0.086,
0.082,0.079,0.076,0.071,0.066,0.059,0.053,0.051,0.052,0.057,0.067,0.081,0.103,
0.129,0.165,0.209,0.252,0.292,0.328,0.363,0.398,0.431,0.459,0.479,0.491,0.494,
0.488,0.475,0.457,0.43,0.397,0.357,0.316,0.285,0.254,0.227,0.206,0.189,0.181,
0.171,0.157,0.151,0.162,0.192,0.239)
One recommended way to handle an even number of points and still achieve precision is to combine Simpson's 1/3 rule with Simpson's 3/8 rule, which can handle an even number of points. Such approaches can be found in (at least one or perhaps more) engineering textbooks on numerical methods.
However, as a practical matter, you can write a code chunk to check the data length and add a single trapezoid at the end, as was suggested in the last comment of the post to which you linked. I wouldn't assume that it is necessarily as precise as combining Simpson's 1/3 and 3/8 rules, but it is probably reasonable for many applications.
I would double-check my code edits below, but this is the basic idea.
AUC <- function(x, h=1){
# AUC function computes the Area Under the Curve of a time serie using
# the Simpson's Rule (numerical method).
# https://link.springer.com/chapter/10.1007/978-1-4612-4974-0_26
# Arguments
# x: (vector) time serie values
# h: (int) temporal resolution of the time serie. default h=1
#jh edit: check for even data length
#and chop off last data point if even
nn = length(x)
if(length(x) %% 2 == 0){
xlast = x[length(x)]
x = x[-length(x)]
}
n = length(x)-1
xValues = seq(from=1, to=n, by=2)
sum <- list()
for(i in 1:length(xValues)){
n_sub <- xValues[[i]]-1
n <- xValues[[i]]
n_add <- xValues[[i]]+1
v1 <- x[[n_sub+1]]
v2 <- x[[n+1]]
v3 <- x[[n_add+1]]
s <- (h/3)*(v1+4*v2+v3)
sum <- append(sum, s)
}
sum <- unlist(sum)
auc <- sum(sum)
##jh edit: add trapezoid for last two data points to result
if(nn %% 2 == 0){
auc <- auc + (x[length(x)] + xlast)/2 * h
}
return(auc)
}
sm = smoothed[-length(smoothed)]
length(sm)
[1] 70
#even data as an example
AUC(sm)
[1] 20.17633
#original odd data
AUC(smoothed)
[1] 20.389